Download ANALOGUES OF THE COMPACT-OPEN TOPOLOGY M. Schroder

ANALOGUES OF THE COMPACT-OPEN TOPOLOGY
M. Schroder
(received 1 December, 1978; revised 10 December, 1979)
Introduction
L of subsets of a set
For any collection
space
Y
space
Y^
X
and any uniform
(indeed, for any uniform convergence space), the function
can be equipped with uniform L-convergence.
Y
given the L-open topology (when
It can also be
is topological), a topology shown
to involve nothing more than the convergence of filters to sets.
idea enables one to define an L-convergence on
space
Y
Y^
This
for any convergence
at all, and then to recapture the L-open topology as a special
case.
X
Now suppose that
is a topological space and that
collection of continuous functions from
finds
X
(i) that if
X
Y .
is compactly generated and
then the compact-open topology coincides on
convergence,
to
X
(ii) that if
C
C
is a
Classically, one
Y
is topological,
with continuous
is locally compact and
Y
is a uniform
space then continuous convergence and the topology of compact convergence
coincide on
C , and
(iii) that compact convergence and the compact-open
C
topology coincide on
Y
when
is a uniform space.
More generally, let
C
consist of all
same filters converge to
f
under both L-convergence and uniform
L-convergence.
functions from
X
Then
(a) each member of
X
L
to
f
in
Y^ , such that the
becomes a compactly generated space, and
is compact,
Y , and
(b)
C
is the set of all continuous
(c) L-convergence, uniform L-convergence
and continuous convergence all coincide on
satisfy certain mild restrictions.
C , provided that
L and
This extension and partial converse
Math. Chronicle 10(1981) 83-98.
83
Y
of (iii) above is matched by similar improvements to (i) and (ii):
the
main problem left is - how much can the restriction be relaxed?
Background
Except for the basic calculus of filters, most tools used later are
briefly introduced in this section:
fuller treatment can be found in
the works cited, results being stated mainly without proof.
1.
Sets and filters.
Q
for the power set of
if
A c Q
on
{i4}
x € Q
and
and
{{as}}
Q , let
For any set
Pow(Q)
A
x
and
respectively.
that
M
If
meets
M
and
W
if
Every function
N
for all
filter on
Lemma 1.
h(K)
P , then
h(A)
in
h(U)
Q , one says
Pow(P) , can be 'extended'
to
Fil(Q)
as follows:
if
Q
r
of filters on
Q
nr
respectively.
A filter belongs
finer than some member of
ultrafilter belongs to
r ,
oT , and
(ii) to
(i) to
oF
(iii) to
oT
=
[A : A € A
for all
84
oT , oT
as soon as it is
provided that every finer
nT
if it is finer than the
fi Iter
nr
A .
are associated
its segmental, Choquet and principal modifications, denoted by
and
is a
finer than
for some ultrafilter U finer than
With each non-void collection
A
U{h(i4) : A € A}.
is defined to be the filter
Under these conditions, an ultrafilter on
is finer than
All other filters
such that
A , B
Fil{P)
to a fl-homomorphism from
is
is non-void.
h : Pow(P) — *■Fil{Q)
li(A U fl) = h(A) fl h(B)
As usual,
0 = Pow(Q)
In particular,
are collections of subsets of
M fl N
stand
are the principal filters based
regarded as a filter, known as the improper filter.
are proper.
Q .
and the set of all filters on
then
Fil(Q)
and
A f f).
Naturally
r
is said to be a segment if
and principal if
T = ttT .
solidification [8] which also leads to
solid instead of Choquet if
2.
r = oT , Choquet if
T = oT
(There is another procedure known as
oT :
so
r
may be called
F = oT .)
Point and set convergence.
One method of formalising the idea of
convergence uses a function, associating with each point, the collection
Thus a (point) convergence
of filters which 'converge' to that point.
y
Q
on
y
is defined to be a function
Q
from
to
Pou(Fil(Q) ) .
Though other more (or less) restrictive definitions have been given
elsewhere (see [5] or [6]), it is assumed here that for all
(^p)
y
(Cj)
x
(®)
is a segment,
belongs to
Similarly, a set convergence on
Pow(Fil(Q))
G
and
H
on
Q
A , B
such that for all
in
Q ,
and
y(x) .
Pow(Q)
is a function from
Pow(Q)
in
and all filters
to
F ,
Q ,
(SQ)
r(4)
(Sj)
A
(S2)
if
is a segment,
belongs to
W
r(>4) , and
is finer than
F € r(>i)
F O G
G € r(B)
and
then
and
H £ r(4 U B).
One often uses more descriptive terms such as
"x
x
is a Y-limit of
G "
or the like, instead of
"F — ► A
under
"F € r(/l)"
F",
and
"G £ y(*)" •
Each set convergence
'restriction' :
F — *■x
T
under
other hand, each convergence
Q
on
r*
y
defines a convergence
iff
F — > {x}
under
F*
r .
by
On the
can be 'extended' to a set convergence
85
y#
F — >- A
as follows:
under
y*
iff
^
is finer than the filter
fl{F : x £ A] , for some choice of the filters
x
F
x
in
y(x) •
However, not every set convergence arises in this way.
take an infinite set
P , define
y(x) = {$,x,U^}
distinct non-trivial ultrafilters, and put
under
F
y* , but in the notation used above,
many
x
in
For example,
where the
in
r(i4)
F = x
x
^'s
are
F —►A
if
for all bar finitely
A .
In all cases though,
y = y**
and
T-convergence implies r**-convergence).
convergence
y
iff
T
is finer than
Further,
r**
r = y*
(that is,
for some
r = r** .
Applying the operators
o
and
it pointwise to the convergence
one obtains its Choquet modification
y
ay
y »
and its principal modification
Choquet or principal when the segments
ny .
Similarly one calls
y(x)
are all Choquet or principal.
Analogous terminology is also used
for set convergences.
Finally, a subset
U
of
Q
is called y -open if
U .
every filter y-converging to some point of
subsets of
than
y .
Q
U
belongs to
The set of all y-open
is a topology, which generates a convergence
Thus one calls
y
topological if
y = xy .
xy
coarser
More information
about topological convergence can be found in [5] and [7] along with
references to some of the original work.
Note though the following
facts.
Lemma 2. (i)
(ii)
The convergence
If U
is y-open and
under y*
(iii)
y
then
is principal iff y A is principal.
F
converges to a subset of
U belongs
In particular, if y
to
F .
is topological then the
y-neighbourhood filter of a set is the coarsest
filter yA-converging to that set.
86
U
3.
Uniform convergence structures.
Dq
and
Q2 .
the diagonal in
collection
(f/Q)
ft
is a segment,
(U.)
D
belongs to
(U^)
FOG
1
(#3)
y
ft
e
Q2 = Q * Q
be a set,
C.ll. Cook and H.R. Fischer [3] called a
Q2
ft of filters on
Again let
a uniform convergence structure if
ft ,
ft if both
belongs to
F
G
and
do,
is symmetric, meaning that it is closed
under converses, and
(i/^)
ft
is closed under composition of relations.
ft a uniform rule if
More generally, one calls
hold:
UQ
to
U3 all
the yet weaker axioms used by some other authors would cause
minor inconvenience later.
The convergence
"F
converges to
in that
y — *- x
x
if
with
ft
derived from a uniform rule
under
under
The properties
meaning that if
w
U^
w
F * x
a)” if
exactly when
belongs to
ft by defining
ft , is symmetric,
uj({/) = w(a:) .
are all stable under Choquet modification,
satisfies
U^
then so does
oft .
Furthermore,
ft is a uniform rule, the convergence derived from
od) , and in particular,
w
is Choquet whenever
oft
ft
coincides
is a Choquet
uniform rule.
Q . A
subset K of Q is said to be compact (or more precisely, y-compact)
if every ultra-filter on Q to which K belongs y-converges to some
point of K , and y itself is called compact if Q is y-compact.
4.
Compactness and regularity.
Let
y
be a convergence on
Only a rather weak regularity axiom is needed here:
Rj i
if no proper filter both y-converges to
87
x
y
is called
and rry-converges to
y,
y — *■x
unless
under
(In [8], a slightly stronger axiom, R^
y •
R^ ^ Choquet convergence
was needed to prove that any compact symmetric
was topological, and hence regular.
Similarly, one can show that any
R^ j Choquet convergence is principal:
compact symmetric
the existence
of compact Hausdorff principal convergences which are not topological
R^ ^
shows that
5.
is strictly weaker than
Multiplication and covers.
One constructs convergences
(W*)
F —
x
and
R
Let
y*6
under
in
U
y*S
G
(M-:)
F —+ x
in
6 (as)
G
all
Clearly
if
and
y
y*6
y
G
in
Q
in
is finer than
y*6
Q .
6(x)
can be
fl{R
y
:y $ G) ,
iff there is a filter
such that
in
be convergences on
G , and
F —► G
under
y*
G
for
G .
is finer than
is principal.
t
under
6
by demanding
for all
F
in
and
y*6
iff filters
y(j/)
so chosen that
for all
y
and
R^ ^ .)
y*6
and equally clearly, they coincide
Elementary properties of
•
were covered in [7],
seems to behave similarly (no hidden significance attaches to the
division sign).
As
was said to be diagonal (for reasons given in [5]
y
and [7]), it makes sense to call
strongly diagonal if
y
This has a close connection with topology, since
y
y = yiy .
is topological iff
it is principal and diagonal [5 ].
For some purposes however, one wants
y
to be diagonal, not every­
where but only on a set or collection of sets.
collection of subsets of
induced by the inclusions
UM
then
G —►x
y(M)(x) = {0,x}
under
y
and
M
Q
and let
M c Q .
y(M)
To be precise, if
and otherwise,
meets
So suppose
G — »- x
G fl x , so that
88
M
is a
be the final convergence
x
under
lies outside
y(M)
x € M € G
iff
for some
M
in
M .
that
One then says that
generates
M
y
covers
M
UM = Q
if
y =
and
filter with a base in
and the same y - limit, and that
M
y = yy(M)
compactly generated if they generate
In
y , and
y .
Convergence in function spaces
P
Let
N
and
P , Q
of
and
Q
R =
be sets, and
R
.
Given subsets
L , M
and
respectively, one defines
N(L)
{/ (ac) : / ( N
=
[L,M]
=
x t L) ,
and
{ / € / ? : /(L) c M }
as usual, and notes that
N(L ) c M
(*)...
Further, for any filters
filter on
Q
based on
9
on
subsets of
iff
R
N c [L,M ] .
and
(T(F) : J1 ( 0
Npw take a set convergence
0 —►/
r
on
F
L : F
iff
6(6) — ► f(L)
on
and
Q
P , let
0(F)
be the
f} .
and a collection
P , and define a convergence on
under
R
under
L
of
by demanding that
r
for all
L
in
L .
Basic properties of this type of convergence are listed below without
proof.
Theorem 3.
(i)
(ii)
In the notation used above3
both
P : r
,
is M-diagonal
y
y - y*y(M) .
or
is locally compact if the y-compact sets cover
y
y(M)
if for each y-convergent filter there is a coarser
or strongly M -diagonal according as
particular,
y
{0} : r and 0 : r are the indiscrete convergences
is pointvise convergencet if
singletons,
89
P is the set of all
K :
(iii)
L : r , if
is coarser than
r
on the other hand, if
(iv)
K c L,
is the closure of
L
under finite unions then
K
K : r = L : r , and
if
i = UL. then a filter converges under L : r
i ^
iff it converges to the same limit under each
(v)
L. : r
(that is,
L : T is the supremum of the
l- :.!*)•
The consistency condition
S2
is needed only in proving (iv) above:
SQ
the other claims hold for set convergences satisfying
If
Lemma 4.
Proof.
r
is principal, so is
Suppose
—►/
under
L : r .
L : r .
be the coarsest filter T-converging to
than
G
Lj
. Then by
(*) ,
S j alone.
and
For each
L
in
L
let
i\>(L)
f ( L ) , so that
G^
is finer
i|/ itself is finer that the filter
6„
j
generated by
{[ L,M] : M t Gl
a filter which clearly converges to
and
/
L ( L) ,
under
L : r , as desired.
It is now only a short step to the compact-open topology.
a convergence
y
respectively.
Then
on
Q , and subsets
[L,Af]
[L,Af] , and that
M
is y-open.
f(L) c M .
By lemma 2,
belongs to
\p .
Theorem 5.
Let
y
is open in
ip — *■f
prove this, suppose that
M
L
M
of
P
and
{£} : y*
if
M
is y-open.
and
(L) : y * , that
under
i|<(L) — *■f(L )
Then
belongs to
be topological.
ip(L)
Then
and so by
L : y*
f
under
Q
To
belongs to
y*
and
(*) , [L,M]
is also topological
being the convergence derived from the L-open topology.
90
Consider
Proof.
i , so that
in
L : y*
By theorem 3,
Y-open.
[£,M]
is open in
L :
if
L
f
for
L
M
is
L and
is in
As the neighbourhood filter of a function
topology is generated by sets of the form
M
{L } : y*
is the supremum of the
in the L-open
f(I>) c M
[L,M] , where
is y-open, it coincides by lemma 2 with the filter
0^,
and
defined in the
proof of lemma 4.
Consider now a uniform rule
Alongside
L : w*
definition [4, §7],
0 — *• /
filter
based on
(0 x
(T x {/})(Z?L )
for
T
in
Q
fl on
whose convergence is
one also finds its uniform counterpart
L : fl
under
f(x)) : g £ T
=
0 , belongs to
fl .
L
iff for each
at .
L : fl .
in
By
L , the
x f L]
and
Even in the simplest case of a metric
space such as the real line, the L-open topology and the topology of
L-convergence do not always coincide, and may indeed be incomparable.
Naturally one asks
(i) when does
L : fl-convergence, and
L : id*
convergence imply
(ii) when does the converse hold?
Partial
answers are given in the next section.
First comparisons
Thoughout this section, let
Q
P , fl be a uniform rule on
from
P
functions
by
Q , and
to
h
and
h(A) = 0(i4)
k
and
0
L
be a collection of subsets of
whose convergence is
be a proper filter on
from
Pow(P)
k(A) =
to
(0 x ^ ( Z ^ )
Fil(Q)
u> , /
p
E = Q .
Then the
Fil(Q2 ) , defined
and
for all
A
in
Pow{P) ,
satisfy the condition laid down in lemma 1 .
Suppose first that
(Ag)
each
0 —►/
f(L)
under
L :
, and that
lies inside an w-compact set,
91
be a function
(Tj)
L has the finite intersection property,
(X 2 )
if
F
G
Proof.
under
ft is Choquet, and
(V2)
u>
is strongly compact-diagonal.
0 —►/
L
Take
in
under
L : ft .
So, let
U
By lemma 1, there is an ultrafilter
By
XQ , its image
K
f(L) .
If
/(ti)
0(U) 3 0 (A/)
R
ft , or
be an ultrafilter
9 (M) — ► f(M)
This means that
0(U) — *- y
y
under
under
under
w
y
in a w-compact set
X^ .
by
u>* , for all
w * aj({A})
R ^ k(U) .
such that
U fl L , then
1/ is the filter based on
and
L
converges to some point
/(V) , a filter converging to
K
belongs to
Further,
M
U fl L .
in
and hence under
w , by
.
But
to
belongs to
k(L) = k(L) .
finer than
72
k(L)
L , and try to show that
V ^ , that every finer ultrafilter does.
by
w , and
w ,
(1^)
Then
under
F fl L , then
is the filter based on
/(G) — *■y
Lemma 6.
L , /(F) — > y
meets
ft
k(U)
is clearly finer than
since both
0 (U)
the still finer filter
and
R
/(U)
also belongs to
On the other hand, suppose that
(X3)
each
0(U) x /(U) , a filter belonging
converge to
f(Q
0 —> /
y
under
w .
ft , as desired.
under
L : ft , and that
is m-compact
(X3) ft is a uniform convergence structure, and
( V
o) is
F ,.i
and Choquet.
92
In short,
Then
Lemma 7.
Proof.
Q — *■f
under
L
Again take
in
L : to* .
L , let
R
h(L) , and "choose" an ultrafilter
y
, some point
f(L)
of
be an ultrafilter finer than
il
L
R ~> h(U) .
such that
f(U ) .
is the to-limit of
k(L)
f(U) * f(U)
belongs to
h(U) x /(U)
= k(U ) o {/(U) x f(U)} , as can be seen by comparing basis
fl , as do
and
k(U ) .
By
Consequently
Further
elements:
T(U)
h(U )
Thus
and
X /([/)
R
y
finer than
f(L )
Though
*
G^ = 0
f(L)
in
h{L)
Zy
if
G
y
Being finer than
y —> x
(because
under
w
to
under
let
to .
In short, every
f(L) .
be the set of all ultrafilters
y , and
is void, in all cases
y
So, suppose
y
y
converges to some point of
G
y
G^ —
be their intersection.
y
under
: y t /(£)) , it converges to
as desired, provided that each
G .
{/})(zy o (f(U) x f(U)) .
x
which w-converge to
h(L ) = fl{G
Further, as
(T
both converge to
ultrafilter finer than
For each
=
G^ — * y
under
R
h(L) , it to-converges to some point
(because
to
j) and so
is
— *■y
Hence
under
u* ,
to .
is proper, and take an ultrafilter
is symmetric).
nw .
f(L)
under
R — *- y
to
finer than
x .
Thus
under
to
(because
to
is
C h oquet).
Altogether, conditions
(K)
fl
to
are equivalent to
is a Choquet uniform convergence structure
whose convergence
to
is
^2 1 anc* stron8ly
compact-diagonal.
One should ask whether
V
forces
generated by a classical uniformity.
93
fl to be principal, and hence
Consider the following example.
Q be the set of all complex numbers in the open right half
Let
plane, together with the origin.
Qn
and let
and
t
T
be the usual topology and uniformity on
n .
for some
U
fl , put
and
t
Q
and
in
fl
Q .
U ? T
iff
and
Clearly a filter w-converges to a non-zero limit
H — *• 0
iff it x-converges to that limit, while
under
n > 0 , set
{x + iy € Q : \y\ < nx) ,
=
To define the uniform rule
Q^2 £ U
For each integer
G H , for some
n .
fl are the inductive limits of
By construction,
w
under
u>
iff
H — *■0
(In more technical language,
T l#n
and
<d
respectively.)
is neither topological nor even principal -
but it is c-embedded, [l, Theorem 33] and hence regular, and it is
strongly diagonal (as one can see after a little calculation).
fl
Moreover,
is a Choquet uniform convergence structure, but clearly not principal.
ft and
In all,
ui
satisfy
V without being "classical".
These results may make better sense when one sees that
X 3 help define a convergence on
gence
y
P .
Q , and a collection
°n
X^ , X2
and
To be precise, consider a conver­
L of subsets of
P
satisfying
,
and let
C
Clearly
C
By
/(F) — ► f(x)
X^
and
and
*3
hold} .
contains all the constant maps.
The convergence
iff
{/ € QP : X2
=
B
under
X 3 , each
every member of
on
C
y
L
P
is defined as follows:
f
for all
in
is 6-compact and
is B-y-continuous.
C , and
94
L
meets
L generates
8 .
under
H —►y
B
F fl x .
Also,
(A function is said to be
continuous if it "preserves convergence", that is,
0 (H) — * g(y) •)
F —> x
implies
The converse is true too, at least if
g
this, suppose
L .
meets
G
F
Then
1/ converges to some point
g(x ) = y
But since
x
under
X2 and
under
y , and
F
F fl L .
U
x
L
in
W
y
V fl L
based on
y
under
is Choquet.
in
g(V) — ► g(x )
and
converges to
y , because
L
Choose a 8-compact set
and the filter
B , the filter
g(G) — *- y
To prove
U o g(G ) , one can "find" an ultrafilter
g{\J fl L) c (j .
such that
under
be the filter based on
Given an ultrafilter
V
is Choquet.
g(T) — * y
is B-y-continuous,
Also, let
y
under
y .
converges to
y .
Hence
g
In all,
(/ fl L .
satisfies
*3 .
A global comparison of
Theorem 8. Let
satisfy
L
functions satisfying
L :
, ft satisfy
X2 and
coincide on
L : w*
L : fl and
X3 .
id*
is now easily made.
V s and
C be the set of
Then the convergences
L : ft and
C .
Second Comparisons
Next let
a
and
Con(a,y)
and let
y
be convergences on
Despite its name, continuous convergence
Con(a,y):
, not just on
under
y
whenever
continuously to
C j outside
Theorem 9.
diagonal.
f
F — >■x
iff
Con(a,y) .
f
Q
and
respectively,
L : y*
0 — *■f
under
con(a,y)
under
a .
can be defined on
con(a,y)
However as
f
con(a,y)
is continuous,
0(F) — *■f(x)
iff
converges
does not satisfy
For more details, see [3], [5] or [l].
Suppose that
Then
P
stand for the set of all a-y-continuous functions.
L generates
and that
a
y
is strongly
is finer than continuous convergence, on
Con(a,y) .
Proof:
If
G
Let
0 — *■f
under
L : y* , and
is the filter based on
F
F fl L , then
95
converge to
/(G) -*■ /(x)
x
under
under
a .
a ,
while
0(F) ^ 0(G)
and
Hence
0(F) — *• f(x)
0(C) —> f(G)
under
under
y* , for all
G
in
y
is merely
F
fl
L .
y -’
y = y , as desired.
From the proof, one can see that this remains true if
strongly compact-diagonal, but the members of
L are cx-compact.
In the
uniform case, the comparison below is well known.
Theorem 10.
cover
a .
Let
ft be a uniform convergence structure on
Then
Q , and
L
L : ft is finer than continuous convergence, on
Con{a,uj) .
K .
Next, consider {^}-convergence, for any a-compact set
By
methods similar to those of the previous section, one can prove that
these are coarser than continuous convergence.
Lemma 11.
Let
K be a-compacts and
y be
Then continuous convergence is finer than
Lemma 12.
Let
K be a-compact, and
continuous convergence is finer than
Theorem 13.
diagonal,
Let
y be
R 1 1 , symmetric and Choquet.
{K}:y*,
on
P
Q .
SI be a Choquet uniform rule.
(if) : ft , on
Then
Con (a, to) .
Rj l J symmetric, Choquet and strongly compact-
SI be a Choquet uniform convergence structures and
K
the
set of all a-compact sets.
(i)
If a
is compactly generated then
continuous convergence, on
(ii)
If a
Theorem 14.
If a
More generally, if
,
then
a(K)
K : ft coincides with
Con(a,w) .
is compactly generated and
continuous convergence,
K : ft and
K
coincides with
Con(a,y) .
is locally compact then
continuous convergences on
K : y*
ft satisfies
K : ai* all coincide, on
V s then
Con (a, to) .
is a family of a-compact sets, satisfying
is a compactly generated convergence finer than
96
a .
Thus if
U
V , then both
satisfies
con(a(K) ,u>) , on
Con(a(K) ,w)
K : ft and
K :
and in particular, on
coincide with
Con(a,y) .
In fact, one can prove theorem 8 using these comparisons with
continuous convergence, theorem 14, and the convergence
8
defined
earlier; the proof is no easier, though.
Several problems remain.
can
R^ j be weakened, say, to
An obvious technical one concerns
R^
V :
(I guess not) or strong compact-
diagonal ity to compact-diagonality (I don't see how)?
Since the "constant function" map embeds
y
continuous convergence can be topological only if
in
y
con(a,y) ,
is.
In cases
discussed by R. Arens and H. Poppe, say, [6, Theorems 2.8, 2.8(a) and
2.9], if
con(a,y)
is topological then
compactly generated.
L-convergence, then
is:
a
is locally compact or
con(a,y)
Similarly, if
what conditions on
a
coincides with
is "often" compactly generated.
a
and
y
The problem
lie behind this "often"?
REFERENCES
1.
E. Binz, Continuous Convergence on
C(X), Lecture Notes in
Mathematics 469, Springer, 1975.
2.
E. Binz, H.H. Keller, Funktionenraume in der Kategorie der
Limesraume, Ann. Acad. Sci. Fenn. Ser A1 383 (1966).
3.
C.H. Cook, H.R. Fischer, On Equicontinuity and Continuous
Convergence, Math. Ann. 159 (1965), 94-104.
4.
C.H. Cook, H.R. Fischer, Uniform Convergence Structures, Math.
Ann. 173 (1967), 290-306.
5.
H.J. Kowalsky, Limesraume und Komplettierung, Math. Nachr. 11
(1954), 143-186.
6.
H. Poppe, Stetige Konvergenz und der Satz von Ascoli und Arzela,
Math. Nachr. 30 (1965), 87-122.
7.
M. Schroder, Adherence operators and a way of multiplying
convergence structures, Math. Chronicle 4 (1976), 148-162.
8.
M. Schroder, Compactness Theorems, in Lecture Notes in Mathematics
540, Springer, 1976, 566-577.
University of Waikato
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